Enigma: the 159 quintillion reasons it was thought unbreakable.

A lot of films have been made about Alan Turing and the cipher device Enigma, with The Imitation Game — Benedict Cumberbatch as Turing — one of the more recent. Why was the Enigma thought to be unbreakable? How many combinations were possible for a ciphertext? Let's check the math.

Rotors

Enigma machines came with five possible rotors, with three placed in the machine at once. That gives 5 × 4 × 3 = 60 combinations.

Rotor starting points

Each rotor has 26 starting positions. For three rotors, that's 26³ = 17,576 combinations.

Plugboard

There are 26 letters to pair up, which gives 26! (26 factorial) arrangements. But the machine was provided with only ten plugboard wires, so 20 of the 26 letters are paired and six are left unused: 26! / 6!. Since the order of the ten pairs doesn't matter, divide by 10!. And since it doesn't matter how the letters within each pair are ordered (ab = ba), divide by 2¹⁰. That gives:

26! / (6! × 10! × 2¹⁰) = 150,738,274,937,250 plugboard combinations.

Total

Rotor combinations × rotor starting points × plugboard combinations ≈ 159 × 10¹⁸ possible configurations.

Now you get why the Enigma was thought to be unbreakable, right? But in life, nothing is impossible.